Elasticity is about stretchiness or responsiveness. If you say demand is elastic, it means it is responsive. What is it responsive to?
We mainly talk about price elasticity of demand; this tells us how much demand changes if price changes. For example, if the price elasticity of demand is -2, this means that an increase in price of 10% would lead to a 20% decrease in quantity demanded. [If the absolute value is greater than 1, we say that demand is price elastic (or sometimes we just say "elastic" if it's obvious that we're talking about price elasticity) because the change in price has brought about a more than proportionate change in quantity demanded.]
However, we also measure income elasticity of demand, which tells us how much demand changes if there's a change in income, and cross elasticity of demand, which tells us how much demand for Good X changes if the price of Good Y changes.
We also measure price elasticity of supply, which measures how much supply changes if price changes. For example, if the price elasticity of supply is 0.4, then a 10% increase in price leads to a 4% increase in supply. This is called a relatively inelastic supply because the increase in price leads to a less than proportionate increase in supply.